Find the value of (2x + 3)² − (2x − 1)² − 6x at x = 1.

Step-by-step solution

1. 1

   Substitute directly as a first pass. With $x=1$,

   $(2+3)^{2}-(2-1)^{2}-6=5^{2}-1^{2}-6=25-1-6=18.$

   That answers the question, but it hides the structure — and structure is what protects against arithmetic slips.

2. 2

   Redo it with the difference of squares. The first two terms have the form $A^{2}-B^{2}=(A+B)(A-B)$ with $A=2x+3$ and $B=2x-1$:

   $A+B=4x+2,\qquad A-B=(2x+3)-(2x-1)=4.$

   The $x$ terms cancel in $A-B$, which is why this factorisation is so effective here.

3. 3

   Simplify the whole expression.

   $(4x+2)(4)-6x=16x+8-6x=10x+8.$

   The original quadratic-looking expression is in fact linear — the $4x^{2}$ terms cancelled.

4. 4

   Evaluate the simplified form.

   $10(1)+8=18,$

   agreeing with the direct substitution.

5. 5

   Check a second point to confirm the simplification. At $x=3$ the original gives $(9)^{2}-(5)^{2}-18=81-25-18=38$, and the simplified form gives $10(3)+8=38$ ✓. Because $10x+8$ matches at two points and both expressions are polynomials of degree at most one after cancellation, they are the same function.